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Theorem isdivrngo 36459
Description: The predicate "is a division ring". (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
Assertion
Ref Expression
isdivrngo (𝐻𝐴 → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))

Proof of Theorem isdivrngo
Dummy variables 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 5110 . . . . 5 (𝐺DivRingOps𝐻 ↔ ⟨𝐺, 𝐻⟩ ∈ DivRingOps)
2 df-drngo 36458 . . . . . . 7 DivRingOps = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ RingOps ∧ (𝑦 ↾ ((ran 𝑥 ∖ {(GId‘𝑥)}) × (ran 𝑥 ∖ {(GId‘𝑥)}))) ∈ GrpOp)}
32relopabiv 5780 . . . . . 6 Rel DivRingOps
43brrelex1i 5692 . . . . 5 (𝐺DivRingOps𝐻𝐺 ∈ V)
51, 4sylbir 234 . . . 4 (⟨𝐺, 𝐻⟩ ∈ DivRingOps → 𝐺 ∈ V)
65anim1i 616 . . 3 ((⟨𝐺, 𝐻⟩ ∈ DivRingOps ∧ 𝐻𝐴) → (𝐺 ∈ V ∧ 𝐻𝐴))
76ancoms 460 . 2 ((𝐻𝐴 ∧ ⟨𝐺, 𝐻⟩ ∈ DivRingOps) → (𝐺 ∈ V ∧ 𝐻𝐴))
8 rngoablo2 36418 . . . . 5 (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ AbelOp)
9 elex 3465 . . . . 5 (𝐺 ∈ AbelOp → 𝐺 ∈ V)
108, 9syl 17 . . . 4 (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ V)
1110ad2antrl 727 . . 3 ((𝐻𝐴 ∧ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)) → 𝐺 ∈ V)
12 simpl 484 . . 3 ((𝐻𝐴 ∧ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)) → 𝐻𝐴)
1311, 12jca 513 . 2 ((𝐻𝐴 ∧ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)) → (𝐺 ∈ V ∧ 𝐻𝐴))
14 df-drngo 36458 . . . 4 DivRingOps = {⟨𝑔, ⟩ ∣ (⟨𝑔, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
1514eleq2i 2826 . . 3 (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ ⟨𝐺, 𝐻⟩ ∈ {⟨𝑔, ⟩ ∣ (⟨𝑔, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)})
16 opeq1 4834 . . . . . 6 (𝑔 = 𝐺 → ⟨𝑔, ⟩ = ⟨𝐺, ⟩)
1716eleq1d 2819 . . . . 5 (𝑔 = 𝐺 → (⟨𝑔, ⟩ ∈ RingOps ↔ ⟨𝐺, ⟩ ∈ RingOps))
18 rneq 5895 . . . . . . . . 9 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
19 fveq2 6846 . . . . . . . . . 10 (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺))
2019sneqd 4602 . . . . . . . . 9 (𝑔 = 𝐺 → {(GId‘𝑔)} = {(GId‘𝐺)})
2118, 20difeq12d 4087 . . . . . . . 8 (𝑔 = 𝐺 → (ran 𝑔 ∖ {(GId‘𝑔)}) = (ran 𝐺 ∖ {(GId‘𝐺)}))
2221sqxpeqd 5669 . . . . . . 7 (𝑔 = 𝐺 → ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)})) = ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)})))
2322reseq2d 5941 . . . . . 6 (𝑔 = 𝐺 → ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) = ( ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))))
2423eleq1d 2819 . . . . 5 (𝑔 = 𝐺 → (( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp ↔ ( ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp))
2517, 24anbi12d 632 . . . 4 (𝑔 = 𝐺 → ((⟨𝑔, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp) ↔ (⟨𝐺, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
26 opeq2 4835 . . . . . 6 ( = 𝐻 → ⟨𝐺, ⟩ = ⟨𝐺, 𝐻⟩)
2726eleq1d 2819 . . . . 5 ( = 𝐻 → (⟨𝐺, ⟩ ∈ RingOps ↔ ⟨𝐺, 𝐻⟩ ∈ RingOps))
28 reseq1 5935 . . . . . 6 ( = 𝐻 → ( ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) = (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))))
2928eleq1d 2819 . . . . 5 ( = 𝐻 → (( ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp ↔ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp))
3027, 29anbi12d 632 . . . 4 ( = 𝐻 → ((⟨𝐺, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp) ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
3125, 30opelopabg 5499 . . 3 ((𝐺 ∈ V ∧ 𝐻𝐴) → (⟨𝐺, 𝐻⟩ ∈ {⟨𝑔, ⟩ ∣ (⟨𝑔, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)} ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
3215, 31bitrid 283 . 2 ((𝐺 ∈ V ∧ 𝐻𝐴) → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
337, 13, 32pm5.21nd 801 1 (𝐻𝐴 → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3447  cdif 3911  {csn 4590  cop 4596   class class class wbr 5109  {copab 5171   × cxp 5635  ran crn 5638  cres 5639  cfv 6500  GrpOpcgr 29480  GIdcgi 29481  AbelOpcablo 29535  RingOpscrngo 36403  DivRingOpscdrng 36457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-1st 7925  df-2nd 7926  df-rngo 36404  df-drngo 36458
This theorem is referenced by:  zrdivrng  36462  isdrngo1  36465
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